Non-planar vibrations of an ideal string against a smooth unilateral obstacle

Abstract

In this paper, we study the various possible motions of a string free to vibrate in two mutually perpendicular planes in the presence of a finite unilateral curved obstacle. We consider an obstacle which is curved only along the direction of the string at rest, and which is located at one of the ends of the string. The nonlinear problem of a non-planar string vibration against an obstacle is investigated using a kinematic numerical model under a number of simplifying assumptions. The complex interaction of the string with rigid obstacle is studied without the interfering effects of wave dissipation and dispersion. Also, it is assumed that no energy is lost due to friction and collision of the string with the obstacle. In this paper, we are especially interested in strings that are excited primarily parallel to surface of the obstacle. The modeling results show that presents of the obstacle changes dynamics of the string motion qualitatively. The conclusions of this studied idealized scenario are relevant to string vibrations in Indian stringed musical instruments like sitar or in Japanese shamisen. These lutes are equipped with finite curved bridges, and their strings are primarily excited parallel to those bridges.